In chapter 1 we have anticipated that it is possible to find turbulent states in nonlinear field theories, whose structure is much simpler than the Navier–Stokes equations. In the following, we shall discuss an important case, the so-called complex Ginzburg–Landau equation, which has been intensively studied in recent years, both theoretically and experimentally. Experimental works include oscillatory chemical reactions [Zaikin and Zhabotinsky 1970, Winfree 1972, 1987, Kuramoto 1984, Ouyang and Flesselles 1996], surface catalysis [Jakubith et al. 1990], multimode lasers [Arecchi et al. 1990, 1991], intracellular waves [Lechleiter 1991], colonies of social amoebae [Gerisch and Hess 1974], and cardiac arrhythmia [Winfree 1987, 1989, Davidenko et al. 1991].
For this system many of the basic mechanisms leading to turbulence have been understood, although clear examples of turbulent states have been very hard to observe experimentally (see however Ouyang and Flesselles ). Certain excitations, namely spiral waves or vortices, play a special role and the study of their properties allows quantitative predictions, e.g. for the onset of turbulence. This is very similar to what one hopes to accomplish for the Navier–Stokes equations by studying ‘coherent structures’.
Most of the chapter will be concerned with the complex Ginzburg–Landau equation in a two-dimensional spatial domain. Here the vortex excitations are point-like and the geometry is easy to visualize. In addition, lots of recent experiments have been performed on surfaces or in shallow reaction dishes, i.e. in essentially two dimensional systems.